It states that every positive integer can be expressed as the sum of at most four squares.
More formally, for every positive integer n there exist non-negative integers a,b,c,d such that n = a2 + b2 + c2 + d2
Adrien-Marie Legendre improved on the theorem in 1798 by stating that a positive integer can be expressed as the sum of at most three squares if and only if it is not of the form (4k)(8l-7). His proof was incomplete, leaving a gap which was later filled by Karl Friedrich Gauss.